Why don't numbers stop? - Reyhane, 7 years old, Tehran, Iran
Here's a game: ask a friend to give you a random number and you give back one that is bigger. Just add "1" to the number they come up with and you're guaranteed to win.
The reason is that numbers go on forever. There is no highest number. But why? As a mathematics professor, I can help you find an answer.
First you need to understand what numbers are and where they come from. You learned about numbers because you could count with them. Early humans had similar needs, whether they needed to count animals killed while hunting or keep track of how many days had passed. That's why they invented numbers.
But then the numbers were quite limited and they had a very simple shape. Often the 'numbers' were just notches on a bone, adding up to a few hundred at most.
When the numbers got bigger
As time passed, people's needs grew. Herds of cattle had to be counted, goods and services traded, and measurements taken on buildings and navigation. This led to the invention of larger numbers and better ways to represent them.
About 5,000 years ago, the Egyptians began using symbols for different numbers, eventually ending up with a symbol for one million. Since they did not typically encounter larger quantities, they used the same last symbol to represent "many."
The Greeks, beginning with Pythagoras, were the first to study numbers for their own sake, rather than viewing them as mere counting instruments. As someone who has written a book about the importance of numbers, I cannot emphasize enough how crucial this step was for humanity.
By 500 BCE, Pythagoras and his disciples had realized not only that the counting numbers - 1, 2, 3 and so on - were endless, but also that they could be used to explain cool things, like the sounds made when you pull a tight string plucks. .
Zero is a crucial number
But there was a problem. Although the Greeks could mentally think of very large numbers, they had difficulty writing them down. This was because they did not know the number 0.
Consider how important zero is when expressing large numbers. You can start with 1 and add more and more zeros at the end to quickly get numbers like a million - 1,000,000, or 1 followed by six zeros - or a billion, with nine zeros, or a trillion, with 12 zeros.
It was not until around 1200 CE that nil, invented centuries earlier in India, came to Europe. This led to the way we write numbers today.
This brief history makes it clear that numbers have evolved over thousands of years. And while the Egyptians didn't have much use for a million, we certainly do. Economists will tell you that government spending is usually measured in millions of dollars.
Furthermore, science has brought us to a point where we need even greater numbers. For example, there are about 100 billion stars in our Milky Way - or 100,000,000,000 - and the number of atoms in our universe can be as high as 1 followed by 82 zeros.
If you find it difficult to imagine such large numbers, don't worry. It's fine to simply treat them as "many," just as the Egyptians treated numbers above a million. These examples point to one reason why the numbers must continue indefinitely. If we had a maximum, some new use or discovery would certainly cause us to exceed it.
Exceptions to the rule
But under certain circumstances, numbers sometimes have a maximum, because people design them that way for a practical purpose.
A good example is clock or clock arithmetic, where we only use the numbers 1 through 12. There is no 13 hours, because after 12 hours we simply go back to 1 hour. If you were to play the 'bigger numbers' game with a friend who plays clock arithmetic, you would lose if he/she chose the number 12.
Since numbers are a human invention, how can we construct them so that they continue indefinitely? Mathematicians began to address this question from the beginning of the 20th century. What they came up with was based on two assumptions: that 0 is the starting number, and if you add 1 to any number, you always get a new number.
These assumptions immediately give us the list of counting numbers: 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3, and so on, a progression that continues indefinitely.
You may wonder why these two lines are assumptions. The reason for the first is that we don't really know how to define the number 0. For example: is '0' the same as 'nothing', and if so, what exactly is meant by 'nothing'?
The second may seem even stranger. After all, we can easily show that adding 1 to 2 gives us the new number 3, just as adding 1 to 2002 gives us the new number 2003.
But notice that we're saying this should be true of any number. We cannot properly verify this for each individual case, as there will be an endless number of cases. As humans who can only perform a limited number of steps, we must be careful when making claims about an endless process. And mathematicians in particular refuse to take anything for granted.
Here then is the answer to why numbers don't end: it's because of the way we define them.
Now the negative numbers
How do the negative numbers -1, -2, -3 and more fit into all of this? Historically, people have been very suspicious of such numbers, because it is difficult to imagine a 'minus one' apple or orange. As late as 1796, mathematics textbooks warned against the use of negatives.
The negatives were created to solve a math problem. The positive numbers are fine when you add them together. But when you start subtracting, they can't handle differences like 1 minus 2 or 2 minus 4. If you want to be able to subtract numbers at your own discretion, you also need negative numbers.
A simple way to create negatives is to imagine all the numbers - 0, 1, 2, 3 and the rest - drawn equally spaced on a straight line. Now imagine a mirror placed at 0. Then define -1 as the reflection of +1 on the line, -2 as the reflection of +2, and so on. This way you get all negative numbers.
As a bonus, you also know that since there are just as many negative numbers as positive numbers, the negative numbers must also go on forever!
Hello, curious children! Do you have a question that you would like an expert to answer? Ask an adult to send your question to [email protected]. Tell us your name, age and the city where you live. And since curiosity knows no age limit - adults, let us know what you're wondering too. We won't be able to answer all questions, but we will do our best. This article is republished from The Conversation, an independent nonprofit organization providing facts and analysis to help you understand our complex world.It was written by: Manil Suri, University of Maryland, Baltimore County.
Read more: Manil Suri does not work for, consult with, own shares in, or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.
